The 20 year-old moves to Chelsea in a permanent deal after impressing in a short trial stint.
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A former Republic of Ireland U19 international, Osagie will be initially registered with Chelsea's U21 team.
The defender originally came through the youth system at Southend United before joining West Ham in 2020. He then left London for Celtic, signing a three-year contract in 2022.
Osagie joined Peterborough United on-loan last season before landing this move to Chelsea as a free agent.